>It wasn't a symbolic algorithm. Who uses words like "biggest" and "what's left over" as an example of a symbolic algorithm?

If you prefer, and algorithm internalized or not that operates on symbols (like 8 % 3 - those are symbols right?)without needing to convert to the pictures first.If you can do the division 8 % 3 to get the result 2 remainder 2 then you've used such an algorithm, whether you can articulate the algorithm itself as a sequence of symbols or not.

>The goal is a mathematical understanding of fractions.

Yes -- what do *you* mean by that? I spelled it out somewhat what I mean by breaking it down -- can you do the same? And no, we don;t need to quibble over every single term used unless the goal is to quibble.

What I got out of your various posts was hat the materials your son is using makes no attempt to proceed to the symbolic level (or whatever *you* want to call it.)That sounds absurd for sure. What grade level?